Find (∂V/∂T) of a Van der Waals Gas using variables?

[qnzbabi91]

Find (∂V/∂T) of a Van der Waals Gas using variables?
Evaluating a Derivative of the van der Waals Equation using the Cyclic Rule


For the van der Waals equation of state:
V-b(P+ a/v^2)=RT

the derivative (∂V/∂T)p is difficult to obtain directly because finding an equation for V in terms of P or T requires solving the cubic equation:
PV^3 - (bP +RT)V^2 + a(V-b)=0

a and b are two parameters that take into account the size of the molecule and the strength of the attractive interaction.

Because P is linear in the van der Waals equation, it should be easier to find the partial derivatives:
(∂P/∂T)v and (∂V/∂P)T = 1/(∂P/∂V)T

needed to utilize the cyclic rule.

Using the cyclic rule, find
(∂V/∂T)P

Express your answer in terms of the parameters, constants, and variables in the van der Waals equation (P,V,R,T,a,and b).

What exactly is (∂V/∂T) at a constant P in terms of P,V, R, T, a, and b? Thanks!

 

[Asim Riaz]

Using the cyclic rule, (∂V/∂T)p = (∂V/∂P)T * (∂P/∂T)vTherefore, (∂V/∂T)p = 1/(∂P/∂V)T * (∂P/∂T)vSubstituting in the van der Waals equation of state yields:(∂V/∂T)p = 1/[(V-b)(RT + a/V^2)] * (-a/RT)Therefore, (∂V/∂T)p = -a/(RTV^2 - baV +ab)